SOLUTION: The one-to-one function f is defined below. =f(x)=x/5x-4 Find f^-1, where f^-1 is the inverse of f . Also state the domain and range of f^-1 in interval notation.

Algebra ->  Inverses -> SOLUTION: The one-to-one function f is defined below. =f(x)=x/5x-4 Find f^-1, where f^-1 is the inverse of f . Also state the domain and range of f^-1 in interval notation.       Log On


   

Question 1110100: The one-to-one function f is defined below.
=f(x)=x/5x-4
Find f^-1, where f^-1 is the inverse of f
.
Also state the domain and range of f^-1 in interval notation.
Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


The domain of f(x) is all values except 4/5.

That means the range of f^-1(x) is all values except 4/5.

To find f^-1(x), switch x and y in the given function and solve for the new y.

x+=+y%2F%285y-4%29
x%285y-4%29+=+y
5xy-4x+=+y
5xy-y+=+4x
%285x-1%29y+=+4x
y+=+4x%2F%285x-1%29

f%5E-1%28x%29+=+%284x%29%2F%285x-1%29

The domain of f^-1(x) is all values except 1/5.

Graphing the function and the inverse confirm these results.

The graph of f(x) (red) has a vertical asymptote at x=4/5 and a horizontal asymptote at y=1/5 (green).

graph%28400%2C400%2C-2%2C2%2C-2%2C2%2Cx%2F%285x-4%29%2C1%2F5%29

The graph of f^-1(x) (red) has a vertical asymptote at x=1/5 and a horizontal asymptote at y = 4/5 (green).

graph%28400%2C400%2C-2%2C2%2C-2%2C2%2C%284x%29%2F%285x-1%29%2C4%2F5%29

ANSWER:
f^-1(x) = 4x/(5x-1)
Domain: x not = 1/5
Range: y not = 4/5